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\author{ {\fontspec{Trebuchet MS}Marcin Chrz\k{a}szcz} (Universit\"{a}t Z\"{u}rich)}
\institute{UZH}
\title[Testing random number generators]{Testing random number generators}
\date{\fixme}


\begin{document}
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\begin{center}
\begin{center}
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		\begin{column}{0.9\textwidth}
			\flushright\fontspec{Trebuchet MS}\bfseries \Huge {Testing random number generators}
		\end{column}
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		  %\includegraphics[width=\textwidth]{SHiP-2}
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	\quad
	\vspace{3em}
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\flushright \vspace{-1.8em} {\fontspec{Trebuchet MS} \Large Marcin Chrząszcz\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}}

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\vspace{1em}
%		\footnotesize\textcolor{gray}{With N. Serra, B. Storaci\\Thanks to the theory support from M. Shaposhnikov, D. Gorbunov}\normalsize\\
\vspace{0.5em}
	\textcolor{normal text.fg!50!Comment}{Monte Carlo methods, \\ 24 March, 2016}
\end{center}
\end{frame}
}


\begin{frame}\frametitle{General methodology}
 \begin{footnotesize}
 \begin{exampleblock}{How to check if we have a good generator?}
 The generator is good if the number sequences that it produces have properties of truly random numbers.\\
 But how to check this!?!
 \end{exampleblock}
 \ARROW Traditional methodology:\\
 Define some properties of random numbers from $\mathcal{U}(0,1)$ and check if in the tests the properties are conserved.\\
 \begin{itemize}
 \item The problem with this approach is the fact that there are infinite number of test like this one would have to do :(
 \item In practice one can only only prove the generator is bad, but not that it's good.
 \item There is no way to guarantee that if the $n$ tests are fulfilled the $n+1$ will not fail!
 \end{itemize}
\ARROW The testing can be only in terms of so-called negative selection.\\
\ARROW By each test our trust in the generator increases our trust in it, but it's not GM cars! There is no guarantee.
 
 \end{footnotesize}

\end{frame}


\begin{frame}\frametitle{General methodology, example}
 \begin{footnotesize}
 
 \ARROW Let's assume we have a generator that has $\mathcal{U}(0,1)$:
 \begin{itemize}
 \item We generate $n$ numbers($n$ is fixed).
 \item From them we calculate a values of test function $T$.
 \item We calculate the $F(T)$ where F is the CDF of the $T$ statistics.
 \item Repeat the procedure $N$ times: $T_1, ..., T_N$ and $F(T_1),...,F(T_N)$.
 \end{itemize}
 \ARROW If the generator is good(hipothesis of $\mathcal{U}(0,1)$ is true the $F(T_1), ..., F(T_N)$ will have the distribution of $\mathcal{U}(0,1)$. One usually quotes the credibility level of a test!\\
 \ARROW There are number of test that the generator can be applied to. in the literature:
\begin{itemize}
\item ''The Art of Computer Programming'', Author	Donald Knuth
\item \texttt{DIEHARD} by G.Marsaglia, \href{http://stat.fsu.edu/pub/diehard/}{\url{stat.fsu.edu/pub/diehard/}}
\end{itemize}
\ARROW {\color{RoyalPurple} Modern approach:}\\
Us the same formalism as is in studies the classical chaotic dynamical systems (same formalism is used in the modern generators).\\
\ARROW The \texttt{RANLUX} generator fulfils the chaotic test and all known classical tests (not surprisingly ;) )


 \end{footnotesize}

\end{frame}



\begin{frame}\frametitle{The $\chi^2$ texts with $\mathcal{U}(0,1)$}
 \begin{footnotesize}
 
 \ARROW The algorithm:
 \begin{itemize}
 \item Divide the $[0,1)$ into $k$ subdivisions:\\ \begin{align*}
 0=a_0<a_1<a_2<a_3<...<a_k=1
 \end{align*}
 \item Let $a_{n_i}={X_1,X_2,..X_n}$ be an series of elements in the interval $[a_{i-1}, a_i)$ (with $n_i$ elements). The $p_i=P({a_{i-1}<X<a_i}) =a_i-a_{i-1}$.
 \item A random variable:
 \begin{align*}
 \chi^2_k=\sum_{i=1}^k \dfrac{(n_i-np_i)^2}{np_i},~~~~n=\sum_{i=1}^kn_i,
 \end{align*}
 had a $\chi^2$ distribution of $k-1$ degrees of freedom.
 \end{itemize}
 \ARROW The above hypothesis verifies if the random numbers are indeed $\mathcal{U}(0,1)$.\\
 \ARROW The $\chi^2$ distribution: $X \in  \mathbb{R}, X>0, N \in \mathbb{N}$:
 \begin{align*}
 \rho(X)=\dfrac{1}{2} \left(\dfrac{X}{2}\right)^{\frac{N}{2}-1} e^{\frac{X}{2} }\left[\Gamma \left(\frac{N}{2}\right)\right]^{-1}~~~~E(X)=N,~~~V(X)=2N
  \end{align*}
 \end{footnotesize}

\end{frame}

\begin{frame}\frametitle{The multi dimension test}
 \begin{tiny}
 
 \ARROW From the obtained numbers we construct an $m$ dimension points:\begin{align*}
 (X_1,X_2,...,X_m),(X_{m+1},...,X_{2m}),...,(X_{(n-1)m+1}, X_{nm}) 
 \end{align*}
 \begin{itemize}
 \item In principle they should have a uniform distribution in an $(0,1)^m$ hipercube.
 \item we divide each edge of the hipercube into $k$ equal subdivisions: $[j-1]/k, j/k),~j=1...k$.
 \item Now: $n_i$ is the number of $m$ dimensional points, which are in the $i$-th hipercube. 
 \item The $\chi^2$ test statistics:
 \begin{align*}
 \chi^2_{k^m-1} = \dfrac{k^m}{n} \sum_{i=1}^{k^m}n_i^2-n,~~~~n=\sum_{i=1}^{k^m} n_i
 \end{align*}
 \end{itemize}
\ARROW Now we construct other points:
\begin{align*}
 (X_1,X_2,...,X_m),(X_2,X_3,...,X_{m+1}),...
  \end{align*}
\begin{itemize}
\item For $N$ random numbers we have $N-m+1$ such numbers.
\item We define the statistics:
\begin{align*}
\psi_0^2=0,~~~~~\psi^2_m=\sum_{i=1}^{k^m}\dfrac{\left[n_1 - \left(N-m+1 \right)/k^m\right]^2}{\left(N-m+1\right)k^m},~~~m=1,2,...
\end{align*}
\item For large $N$ the $(\psi^2_m-\psi^2_{m-1})$ has a $\chi^2$ distribution with $k^m-k^{m-1}$ degrees of freedom.
\end{itemize}
 \end{tiny}

\end{frame}


\begin{frame}\frametitle{Overlaping-pairs-sparse-occupancy}
 \begin{footnotesize}
\ARROW The OPSO (G.Marsaglia 1984)is an analysis of pairs obtained from random number generator.\\
\begin{exampleblock}{}
$X_1,X_2,...,X_n$ - $n$ random numbers obtained from generator. From each number we take $b$ bits from which we construct a second series: $I_1, I_2,...,I_n$, where $I_j \in \left[0,1,...,2^b-1 \right]$.\end{exampleblock}
\ARROW Next we create the pair series:
\begin{align*}
(I_1,I_2),(I_2,I_3),...(I_{n-1}, I_n)
\end{align*}
\ARROW $Y$ - number of pairs from : ${ (i,j):i,j=0,1,...,2^b-1}$, which DIDN'T occur in the above series.
\begin{center}
\includegraphics[width=0.35\textwidth]{images/test1.png}
\end{center}
\ARROW This kind of test can be exteded to triple-pairs, and quadro-pairs.\\
\ARROW See DIEHARD G.Marsgalia 1993 \href{http://stat.fsu.edu/pub/diehard/}{http://stat.fsu.edu/pub/diehard/} 

 \end{footnotesize}
 \end{frame}

\begin{frame}\frametitle{Kolmogorov - Smirnov}
 \begin{footnotesize}
\ARROW The K-S test is used to check if a Random variable has pdf of a distribution $F$. The test is based on the difference between the two distributions:
\begin{align*}
D_n=\sup_{-\infty < x<\infty}  \vert F_n(x) -F(x) \vert,~~~~~F_n =\dfrac{1}{n}\sum_{j=1}^n \Theta(x-X_j).
\end{align*}
\ARROW If the random generator is from the $F$ distribution then the $D_n \to 0$ with the probability 1.\\ \ARROW Large values of $D_n$ exclude the generator.
\ARROW The critical values of the test $D_n(\alpha)$ can be find in the mathematical tables for every $\alpha$:
\begin{align*}
\mathcal{P}[D_n<D_n(\alpha)]=\alpha
\end{align*}
\ARROW They do not depend on the $F$ function.\\
\ARROW For the $\mathcal{U}(0,1)$:
\begin{align*}
F(x)=x,~0<x<1
\end{align*}


 \end{footnotesize}
 \end{frame}

\begin{frame}\frametitle{Kolmogorov - Smirnov in practice}
 \begin{footnotesize}

\begin{exampleblock}{ Take note:}
Empirical CDF of $F_n$ is a step function and $\sup_{-\infty < x<\infty} \vert F_n(x) -F(x) \vert$ is achieved only in one point!
\end{exampleblock}
\ARROW In practice one should sort the numbers: $X_1,..., X_n$ and calculate the following:
\begin{align*}
D_n^{+}=\max_{1 \leq i\leq n} \left(\dfrac{i}{n} - F(X_{i:n}) \right),~~~~D_n^{-}=\max_{1\leq i\leq n} \left(F(X_{i:n})- \dfrac{i-1}{n}\right)\\
D_n=\max\lbrace D_n^+, D_n^- \rbrace
\end{align*}
where $X_{i:n}$ is so-called position statistic: $X_{1:n}, X_{2:n},..., X_{i:n}$.\\
\ARROW The statistic $D_n$ asymptotically (in practice $n \geqslant 80$ ) is approaching the $\lambda$-Kolomogorows cdf:
\begin{align*}
\lim_{n \to \infty} \mathcal{P} \lbrace\sqrt{n}D_n  \leqslant t \rbrace =K(t)= \sum_{j=-\infty}^{\infty} (-1)^j e^{-2j^2 t^2},~t>0
\end{align*}
for which the critical values $\lambda_{\alpha}(\mathcal{P}\lbrace\sqrt{n}D_n\rbrace ) >\lambda_{\alpha}$ can be found in the mathematical tables.\\
\ARROW Commonly the $\lambda_{0.1}=1.224$, $\lambda_{0.05}=1.358$, $\lambda_{0.01}=1.628$ are used.



 \end{footnotesize}
\end{frame}

\begin{frame}\frametitle{Statistic distributions test- sum test}
 \begin{footnotesize}
\ARROW The $h$ function has the form:
\begin{align*}
y=x_1+x_2+x_3...x_m.
\end{align*}
\ARROW the random variables form the new pdf:
\begin{align*}
g_m(y)=\begin{cases}
\dfrac{1}{m-1} \left[ y^{m-1} - {m \choose 1}(y-1)^{m-1} +{m \choose 2}(y-2)^{m-1}-..   \right]~~&{\rm for~}  0\leq y\leq m,\\
0~~&{\rm else}
\end{cases}
\end{align*}
where you stop when $y-m$ is negative.
\ARROW For $m=2$ we have the triangle pdf:
\begin{align*}
g_2(y) =\begin{cases}
y,~{\rm for}~0 \leq y \leq 1\\
2-y,~{\rm for}~0 \leq y \leq 1\\
\end{cases}
\end{align*}
\ARROW For $m=3$ we have the triangle pdf:
\begin{align*}
g_3(y) =\begin{cases}
\dfrac{1}{2} y^2,~{\rm for}~0 \leq y \leq 1\\
\dfrac{1}{2} \left[y^2-3(y-1)^2\right],~{\rm for}~1 \leq y \leq 2\\
\dfrac{1}{2} \left[y^2-3(y-1)^2 3(y-2)^2  \right],~{\rm for}~2 \leq y \leq 3\\
\end{cases}
\end{align*}
\ARROW For large $m$ the $g_m$ approaches the normal distribution.

 \end{footnotesize}
\end{frame}

\begin{frame}\frametitle{Statistic distributions test- $d^2$}
 \begin{footnotesize}
\ARROW for $m=4$ we define the $h$:
 \begin{align*}
 y=(x_1-x_3)^2 +(x_2-X_4)^2
 \end{align*}
aka the square distance between $(x_1,x_2)$ and $(x_3,x_4)$.\\
\ARROW If the $X_1$, $X_2$, $X_3$, $X_4$ are from $\mathcal{U}(0,1)$ then:
\begin{align*}
d^2 = (X_1-X_3)^2+(X_2-X_4)^2
\end{align*}
had a pdf given by the following formula:
\begin{align*}
\mathcal{P}(d^2-y) =\begin{cases}
\pi y - \dfrac{8}{3}y^{\dfrac{3}{2}} + \dfrac{1}{2} y^2 ~~ &{\rm for }~0 \leq y \leq1 \\
-\dfrac{1}{2} y^2-4 {\rm arcsec}  (y^{\dfrac{1}{2}})~~ &{\rm for}~ 1 \leq y \leq2 \\
\end{cases}
\end{align*}
\ARROW Test is to check if the generated numbers have the aforementioned distribution.

 \end{footnotesize}
\end{frame}


\begin{frame}\frametitle{Statistic distributions test- pair distance}
 \begin{footnotesize}
\ARROW Generate $n$ points from $(0,1)^m$. We take ${ n \choose 2}$ pairs of points and we calculate the distance between them.\\
\ARROW If $D$ is the smallest distance between the pairs $\longmapsto$ for the $\mathcal{U}(0,1)^m$ the $T=n^2D^m/2$ has the exponential distribution with the mean $1/V_m$, where $V_m$ is the hiper volume of the unite ball.\\
\ARROW In Patrice:\\
\begin{itemize}
\item We generate $Nn$ points in the hipercube $(0,1)^m$, getting $N$ points in the $T$ statistics.
\item We compare the empirical distribution $T$ with the exponential distribution.
\item WARNING: the $N,n,m$ need to be choose smartly for the test to make sense.
\end{itemize}
\ARROW Linear generators usually fail this test!
 \end{footnotesize}
\end{frame}



\begin{frame}\frametitle{Statistic distributions test- series test}
 \begin{footnotesize}
\ARROW Lets assume our numbers are generated with a CDF $F$. The values of $F$ we divide in two separated sub-samples: $A$ and $B$. \\
\ARROW Furthermore we define the new variables $Y$ such as:
\begin{align*}
Y=\begin{cases}
=a X ~\in A\\
=b X~ \in B
\end{cases}
\end{align*}
\ARROW The random number sequence we transform the $X_1,X_2,X_3,...,X_n$ into $Y_1,Y_2,Y_3,...,Y_n$. \\
\ARROW Next we make series: For example the $a,a,b,a,a, b,b,b, a$ will be grouped into $aa$, $b$, $aa$, $bbb$, $a$.\\
\ARROW Let $n_a$ be number of $a$ symbols in $Y_1,Y_2,Y_3,...,Y_n$. $n_b=N-n_a$. \\
\ARROW Distribution of number of series ($R$) is given by the equation:
\begin{align*}
\mathcal{P}(R=r, n_a,n_b)=\begin{cases}
2 {n_a- 1 \choose k-1}{n_b-1 \choose k-1}/{N \choose n_a}~{\rm if}~r=2k\\
[{n_a- 1 \choose k}{n_b-1 \choose k-1} +{n_a- 1 \choose k-1}{n_b-1 \choose k} ]    /{N \choose n_a}~{\rm if}~r=2k+1
\end{cases}
\end{align*}

 \end{footnotesize}
\end{frame}



\begin{frame}\frametitle{Statistic distributions test- poker test}
 \begin{footnotesize}
\ARROW The values of $X$ random variable we divide into $k$ identical sub samples:
\begin{align*}
0<a_1<...<a_k=1
\end{align*}
\ARROW For $X_1,X_2,...,X_n$ from $\mathcal{U}(0,1)$:
\begin{align*}
\mathcal{P}(a_{i-1}<X_j<a_i)=\dfrac{1}{k}.
\end{align*}
\ARROW We create the new variables $Y_1$ accordingly:
\begin{align*}
Y_j=i~{\rm if}~X_j\in(a_{i-1},a_i),~i=0,1,...k-1
\end{align*}
\ARROW Now we create ''the fives'':
\begin{align*}
(Y_1,Y_2,Y_3,Y_4,Y_5),(Y_6,...
\end{align*}
\ARROW There are couple of types of fives:
\begin{itemize}
\item[aabcd] pair
\item[aaabc] three
\item[aaaab] four
\item[aaaaa] five
\end{itemize}

 \end{footnotesize}
\end{frame}

\begin{frame}\frametitle{Statistic distributions test- poker test}
 \begin{footnotesize}
\ARROW If the variables are independent then we can calculate the probability:
\begin{center}
\includegraphics[width=0.7\textwidth]{images/poker.png}
\end{center}
\ARROW In practice people choose: $k=2,8,10$\\
\ARROW The agreemnt of the distribution of different types of fives is check using the $\chi^2$ test.

 \end{footnotesize}
\end{frame}


\begin{frame}\frametitle{Conclusions}
 \begin{small}
\ARROW There are infinite number of tests one can invent for the testing of the generators.\\
\ARROW All of the tests are in the same taste: invent a problem where you know the analytic solution, solve the problem and compare the results.\\
\ARROW Homework: Use one of the previously implemented random number generator and :
\begin{itemize}
\item E5.1 Test them with chi-square test k=10.
\item E5.2 Kolomorov-smirnon.
\item E5.3 Multidimensional test.
\end{itemize} 
 \end{small}
\end{frame}

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